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A308650 Number of permutations of [2n] with n components.

%I A308650 #18 Aug 31 2021 08:08:33
%S A308650 1,1,7,58,531,5226,54598,601924,6985987,85328266,1097775922,
%T A308650 14897635468,213581648046,3238686925956,51972937713900,
%U A308650 882473430354888,15839021166164451,300037212548146890,5986554523174314490,125537613562829696828,2760474045847159393466
%N A308650 Number of permutations of [2n] with n components.
%H A308650 Alois P. Heinz, <a href="/A308650/b308650.txt">Table of n, a(n) for n = 0..400</a>
%F A308650 a(n) = [x^(2n)] (1-1/Sum_{j=0..2n} j!*x^j)^n.
%F A308650 a(n) = A059438(2n,n) = A085771(2n,n) = A263484(2n,n).
%F A308650 a(n) is odd <=> n in { A131577 }.
%F A308650 a(n) ~ sqrt(2*Pi) * n^(n + 5/2) / exp(n-1). - _Vaclav Kotesovec_, Jun 25 2019
%e A308650 a(2) = 7: 1|342, 1|423, 1|432, 21|43, 231|4, 312|4, 321|4.
%e A308650 a(3) = 58: 1|2|4563, 1|2|4635, 1|2|4653, 1|2|5364, ..., 4213|5|6, 4231|5|6, 4312|5|6, 4321|5|6.
%t A308650 p[n_] := p[n] = n! - Sum[k!*p[n - k], {k, 1, n - 1}];
%t A308650 (* T is A059438 *)
%t A308650 T[n_, k_] /; n < k = 0;
%t A308650 T[n_, 1] := p[n];
%t A308650 T[n_, k_] /; n >= k := T[n, k] = Sum[T[n - j, k - 1]*p[j], {j, 1, n}];
%t A308650 a[n_] := T[2n, n];
%t A308650 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Aug 31 2021 *)
%Y A308650 Cf. A059438, A085771, A131577, A263484.
%K A308650 nonn
%O A308650 0,3
%A A308650 _Alois P. Heinz_, Jun 13 2019